5.3 Methodology
5.3.5 Sensitivity of the result to the decadal segmentation
As described in Sect 5.3.1, hSNv2i was calculated by averaging annual SNv2 over every consecutive 10 years starting from 1700 (i.e., over the periods 1700–1709, 1710–1719, ... and so on), with a 0-year lag from 1700. Also, Sect. 5.3.4 uses these hSNv2i as decadal inputs and the four linear relationships obtained therefrom, to simulate the four SC parameters. In reality, however, the cosmogenic isotope signals are sampled over any arbitrary ten years. Therefore, we now test how sensitive our results are to the choice of the decadal intervals, over which the averaging is done.
We now repeat the procedure as in Sect5.3.1, but this time starting with the averaging one year later (i.e., 1701–1710, 1711–1720, ... and so on). This procedure is repeated 10 times until we have a total of 10 series, with the first 10-year averaging period of the 10th series being 1709–1718. These 10 decadally-averaged SN series are denoted as Hi, where the subscriptiindicates the lag 0 – 9 in years.
The values in each Hi series are then compared with the four SC parameters, same procedure as described in Sect. 5.3.1and shown in Fig. 5.2. As a result, we obtain 10 sets of linear relationships with 10 lags. A set of four linear relationships is noted as Kj, where the subscript jindicates the lag 0 – 9 in years. The correlation coefficients,Rc, of the four SC parameters a functions of lag (j) are shown in four panels in Fig. 5.5. It can be seen that all the correlation coefficients are rather stable.
Finally, since it is not clear what the correct lag is for a given data set (i.e., which correct Kj to use), we also need to test the Kjwith the hSNv2iwith ten possible lags (Hi).
Therefore, there are in total hundred possible combinations (Hi=0...9,Kj=0...9) to simulate the SC parameters and we examine the stability of all these 100 combinations.
Following the whole simulation procedure and these 100 combinations, we simulate 100 SN series with annual cadence, which are compared to the SNv2. Figure5.6shows the correlation coefficients between these 100 simulations and the SNv2over the whole period of time (1700 – 2017). The x-axis represents the number of the (Hi,Kj) combinations in a two-digit format. For instance, 63 stands for the combination (H6,K3). The mean Rc value of these 100 simulated results is 0.59. This relatively low correlation is attributed to the difficulty of recovering the precise SC phases over the whole period.
Since the overall solar activity, the amplitude of the solar cycle variation, and the annual temporal resolution are the most important ingredients for climate models, a few years of offset in the phase is not fatal. Hence, instead of using correlation coefficient of the whole time series, we use another quantity, ‘amplitude’, which is defined as the difference between the SN maximum and minimum of each cycle (∆S). We compare the
∆S of the simulated SCs with that of the closest observed SC. The correlation coefficients of ∆S of the hundred combinations are shown in Fig. 5.7. The x-axis represents the (Hi,Kj) combination and the meanRc,∆S value is 0.84. Again, this results indicate that the SC can be simulated stably well regardless of the unknown lagsiand j.
Therefore, for the robustness, we combine all the data points of the four linear rela-tionships from each 1-year lag test, as shown in Fig. 5.8, and take the sets of regression coefficients to simulate the SCs parameters from thehSNv2ivalues:
Figure 5.5: Pearson correlation coefficients, Rc, between four SC parameters: (a) Smax, (b)Smin, (c) Lcyc, (d) Lrise, and the hSNv2i. The x-axis indicates the lag j in years from 1700. TheRcaveraged over all 10 values of jis given in each panel.
Figure 5.6: Pearson correlation coefficients between the reconstructed SC and the SNv2. The x-axis represents the 100 combinations (Hi,Kj) from the ten relationships and decadally-averaged SN with offsets (j).
5.3 Methodology
Figure 5.7: Correlation coefficients of∆S (Smax−Smin) between the reconstructed SC and the measured SNv2. The x-axis represents the 100 combinations, (Hi,Kj), from the linear relationships and decadally-averaged SN with offsets (j).
Lrise,c =−(0.07±0.01)× hSNv2iD+(7.7±0.5) (5.2) Lcyc,c−1 =−(0.08±0.01)× hSNv2iD+(15.1±0.6) (5.3) Smax,c =(1.89±0.13)× hSNv2iD+(8.6±6.9) (5.4) Smin,c =(0.22±0.03)× hSNv2iD−(4.1±1.2) (5.5) The SC simulation based on thehSNv2iand this set of regression coefficients (Eqs. 5.2 –5.5) are shown in Fig.5.9. It can be seen that both the overall comparison (Rc=0.7) and the overall amplitude (Rc,∆S=0.91) are in good agreement with the observed SNv2, and are also significant better than those of the average of 100 combinations (Figs.5.6and5.7).
Furthermore, as mentioned in Sect. 5.3.1that we take the GSN series constructed by Chatzistergos et al.(2017), RCH, to estimate the uncertainty of the results. Therefore, we also do the same statistical study (not shown) based on theRCH series. The set of four relationships obtained based onhRCHiseries considering all 10 1-year lags are:
Lrise,c =−(0.08±0.01)× hRCHiD+(8.6±0.67) (5.6) Lcyc,c−1= −(0.09±0.02)× hRCHiD+(16.0±0.9) (5.7)
Smax,c = (1.8±0.2)× hRCHiD+(1.5±4) (5.8)
Smin,c =(0.56±0.06)× hRCHiD−(15.7±3.5) (5.9) The SC reconstruction based on thehRCHiand the relationships Eqs. (5.6) – (5.9) are present in Fig.5.10. The regression coefficients obtained based onRCHseries are largely close to that based on SNv2series, besides the coefficients in the simulatingSmin,c. This is mainly attributed to the issue discussed in (Chatzistergos et al. 2017) that they might have overestimated the level of solar minima, and therefore the slope obtained here is higher than that obtained from SNv2series. In general, the two sets of relationships based on both,
D
D D
D
Figure 5.8: Same as Fig. 5.2, but this time considering all the data points from the relationships obtained with 10 1-year lag tests: (a)Smax. (b)Smin. (c) Lcyc. (d)Lrise. The solid lines are the best OLS-bisector fits with 3-σuncertainty shown by the dashed lines.
5.4 Results
Figure 5.9: Same as Fig.5.4, but reconstructed based on the final relationships Eqs. (5.2) – (5.5) from thehSNv2iseries, H0.
SNv2andRCHseries, are close enough (within each other error range) and the simulation results show that this statistical approach is rather robust for simulating the SCs on longer time scales.
5.4 Results
5.4.1 Simulation of solar cycles over the last 9 millennia
We now use the approach described in Sect. 5.3to simulated the SCs over the past 9000 years for the three sets of decadally-averaged SN derived from the cosmogenic isotopes (Sect.5.2).
As shown in Sect.5.3.1, we need two input quantities for this: the cycle length,Lcyc,f, and the end time, Tend,f, of the last cycle of the series. Two of three isotope-based SN
Figure 5.10: Same as Fig. 5.9, but reconstructed based on the final relationships Eqs.
(5.6) – (5.9) withhRCHiseries, H0.
series,14C-based U16-14C and the multi-isotope composite Wu18, have the last data point in 1895. The two initial values can therefore be determined from the directly measured SN. In this study, the values of Lcyc,f andTend,f for these two series are 11.09 and 1902, respectively.
However, the last data point of U16-10Be series (Usoskin et al. 2016a) ends in 1645, which is the beginning of the Maunder minimum and the sunspot was typically zero. The-refore, determiningLcyc,f andTend,f is not straightforward. Some studies suggested that the Sun may have kept the quasi 11-year cycles during the Maunder minimum although the magnetic activity was not strong enough to form observable sunspots (Beer et al. 1998;
Beer 2000a;Fligge et al. 1999;Usoskin et al. 2001;Miyahara et al. 2004;Owens et al.
2012). Some other studies have proposed that the solar dynamo had a different dynamo mode from the present state (Schmitt et al. 1996;Moss et al. 2008; Choudhuri and Ka-rak 2012;Usoskin et al. 2014;Käpylä et al. 2016), whileCameron and Schüssler(2017) recently demonstrated, however, that a different dynamo mode is not required to explain
5.4 Results the presence and statistics of grand minima. In this study, for the sake of simplicity, we assume that the Sun kept its regular dynamo during the Maunder minimum, i.e., it had a regular cycle length of roughly 11 years (Lcyc,f = 11). In addition, we take 1655 as the solar activity minimum (Tend,f) after the last data point of U16-10Be, by assuming the first observed cycle at the end of the Maunder minimum began around 1699 – 1700.
For each isotope input (U16-14C, U16-10Be and Wu18), we simulate the SCs with both SNv2-based andRCH-based relationships, which are taken as upper/lower bounds of the result. The difference between the two simulations is then considered as the uncer-tainty range. Figure5.11 shows the results of the simulated SCs based on the three data sets: (a) U16-14C, (b) U16-10Be and (c) Wu18, over the period 1200 – 1900 AD, with their uncertainty shown as shaded areas. The original decadally-averaged SN values of each series are shown by the thick dotted lines. The simulated series in the early epoch, over the period 6800BC– 6100BC, based on the three data sets are shown in Fig.5.12.
The uncertainty range values of the simulations based on the three cosmogenic isotope data sets are shown in Fig. 5.13. The annual values for U16-14C, U16-10Be and Wu18 series are shown in light blue, gray and light red, respectively. The 100-year running averages are represented by dark thick curves with the same colours. Additionally, it is seen that the uncertainty values of the Wu18-based simulation are in overall within 10 sunspot numbers, and are smaller than the simulations based on the other two individual isotope series.
These three simulated SN series with quasi 11-year SCs have an annual resolution and are further employed in the semi-empirical model to reconstruct the solar irradiance on millennial time scales.
5.4.2 Reconstruction of solar irradiance on millennial time scale with solar cycles
To reconstruct the solar total and spectral irradiance, we use two SATIRE (Spectral And Total Irradiance REconstruction) versions, SATIRE-T (Krivova et al. 2007, 2010) and SATIRE-M (Vieira et al. 2011;Wu et al. 2018a). The SATIRE models have been descri-bed in details in many publications, therefore, we only briefly outline the main features of the model here.
SATIRE is a family of semi-empirical models to reconstruct TSI and SSI over wa-velength range 115 – 160 000 nm on different time scales using the most relevant solar activity datasets available at a given time. The model prescribes solar variability on time scales longer than a day to the evolution and the spatial distribution of the solar surface magnetic features. The SATIRE-S version (‘S’ for Satellite, Krivova et al. 2003; Ball et al. 2012;Yeo et al. 2014b,2015) uses spatially-resolved full-disc intensity images and magnetograms to determine the sizes and the distributions of the magnetic features on the solar surface and reproduces the measured solar irradiance variability with high accuracy (95%). Back to the Maunder minimum, the SATIRE-T version (‘T’ for telescope,Krivova et al. 2007,2010) uses directly observed SN to infer the solar surface magnetisms from a rather simple physical model represented by a set of ordinary differential equations. On time scales of millennia, the SATIRE-M version (‘M’ for millennia Vieira and Solanki 2010; Vieira et al. 2011;Wu et al. 2018a) uses decadally-resolved cosmogenic radionu-clides in the natural archives as indirect proxies of solar activity. Both, SATIRE-T and
Figure 5.11: Simulated SCs from cosmogenic isotopes between 1200 – 1900 AD. (a) U16-14C. (b) U16-10Be. (c) Wu18. The thick curves with dots are the original decadally-averaged SN. The simulations with SNv2-based and the RCH-based relationships serve as the upper and lower uncertainties in all three series, shown as shaded area.
5.4 Results
Figure 5.12: Same as Fig.5.11. Simulated SCs from three cosmogenic isotope-based SN series between 6800 – 6100BC.
Figure 5.13: Annual values of the uncertainty of the U16-14C (light blue), U16-10Be (gray), and Wu18 dataset-based (light red) over period 6755BC– 1900AD. The 100-year running averages of these uncertainties are shown by the darker thick curves of the same colours.
SATIRE-M, employ computations of the evolution of solar total and open magnetic flux according toSolanki et al.(2000,2002b);Vieira and Solanki(2010).
We refer the reader to paper I for the details on the TSI/SSI reconstruction based on the observed SN with the SATIRE-T model, and the decadal TSI/SSI reconstructions based on the three isotope-based SN with the SATIRE-M model.
The TSI (integral over 115–160 000 nm) reconstructions based on the three isotope data sets are shown in Fig.5.14. The panels from top to down are (a) 14C, (b) U16-10Be and (c) Wu18. The decadal TSI reconstruction with SATIRE-M model are shown as black dotted curves in three panels. Each dot represents a decadal value. Here we improve upon the reconstructions presented in Paper I by starting from annual SN time series reconstructed in Sect. 5.4.1. The presence of the SC and their yearly resolution allows us to employ the SATIRE-T model (Krivova et al. 2007, 2010;Wu et al. 2018a) directly, instead of using the decadal relationship introduced in the SATIRE-M. The TSI reconstructed from these three simulated SCs are also shown in Fig. 5.14: (a) U16-14C, (b) U16-10Be and (c) Wu18. Same as Fig. 5.11, the gray shaded areas represent the difference of the reconstructions with the SNv2-based and the RCH-based relationships.
Since the U16-10Be data set ends in 1645, the TSI reconstruction does not overlap with that based on the directly observed SN (Fig. 5.14b). Therefore, the step (3) which is for adjusting the phase, cannot be applied for this reconstruction. For comparison, the 361-day running mean TSI reconstruction based on the directly observed SN,hSNv2i, using the SATIRE-T model shown in red in all three panels.
Because the multi-isotope composite Wu18 accounts for the temporal discrepancies between one global14C and six regional10Be series, it averages out many of the systematic effects that individual series suffer from. Hence, it is more robust in representing the solar activity and we recommend the TSI reconstructed from it for the use in paleo-climate
5.4 Results
Figure 5.14: TSI reconstructions with SCs from cosmogenic isotopes between 1200 – 1900AD using reconstructed SC in Fig. 5.11. (a) U16-14C. (b) U16-10Be. (c) Wu18.
The thick curves with dots are the TSI reconstructions with decadally-averaged SN. The RCH-based relationships serve as uncertainty in all three series, indicated by the shaded area. 361-day running average TSI reconstruction based on SNv2is shown in red.
Figure 5.15: TSI reconstruction based on the Wu18 SN series (thick black curve) between 1200 – 1900AD. The upper and lower values among the three reconstructions in Fig.5.11 at any given time is taken as the uncertainty in this figure (shaded area). 361-day running average TSI reconstruction based on SNv2is shown in red.
models, as shown by the thick black curve in Fig. 5.15. For the uncertainty, we take the upper and lower values among all three TSI reconstructions (Fig.5.14) at any given time, shown as shaded area in Fig.5.15.
The final, recommended TSI and SSI series is obtained by combining the TSI/SSI based on the yearly observed SN since 1700 with the irradiance deduced from yearly isotope-based reconstructed SN over the Holocene.
5.5 Summary
The concentration of cosmogenic isotopes, in particular14C and10Be, in natural archives is a commonly-used indirect proxy of solar activity for times before telescopic measure-ments of sunspot number started. Various studies have used14C and10Be, to reconstruct solar activity variation over millennia (e.g., Beer et al. 1988, 2012;Solanki et al. 2004;
Steinhilber et al. 2012;Usoskin 2017;Wu et al. 2018b). However, the temporal resolution of those isotope data is usually low, with only decadally-averaged values available. Thus the information on the 11-year solar activity cycle is averaged out. Because the cyclic va-riation is important for paleo-climate modelling that aims to assess the influence of solar irradiance on the global climate, it is necessary to reconstruct the solar cycles prior to the telescope era.
In this work, we consider four main SC characteristics (the values of solar maxima and solar minima, the rise time and the cycle length) of a directly measured sunspot number series. We used the International sunspot number by (WDC-SILSO,Clette et al. 2014) as a main input record. The new SN series by Chatzistergos et al.(2017) constructed with a non-linear non-parametric method is considered to access the uncertainty range.
5.5 Summary We determine linear relationships between each of these four cycle characteristics and decadally-averaged sunspot number.
Nevertheless, this statistical approach can only gives the timings of the solar activity minima and maxima. In order to obtain a reasonably realistic SC shape, we introduced a first non-parametric fit function, which is inspired byVolobuev(2009). This new non-parametric function describes the general solar cycle shape between both the rising and decaying phase. If the SC characters are known, this function describes the general SC shape with an excellent similarity (Rc =0.96) to the observed SN over the last 300 years.
We then use the four new relationships and the newly derived function describing cycle shape to reconstruct solar cycles from the decadally averaged SN records, initially from artificially averaged directly measured SN.
The TSI reconstructions based on the cosmogenic isotopes with this statistical appro-ach show a good agreement (within uncertainty range) with that based on the observed SN, in both amplitude and the overall trend. However, the errors in the cycle phases accu-mulate and can lead to an offset of a few years between observed and reconstructed solar cycles.
Finally, using the approach developed and tested here, we reconstruct the yearly SN from the decadally-averaged isotope-based SN over the last 9 millennia. This simulated SN with a yearly cadence is then employed as a proxy of solar activity to reconstruct the TSI/SSI with the SATIRE-T model. This TSI reconstructed from the SN with a simulated SC matches well with the TSI reconstruction based on directly observed SN.
6 Summary and outlook
不 積跬步
，無以至 千里
，不 積 小流，無以 成 江海。
–
筍子
h勸學
i In this chapter, we will first summarize the work done in this thesis, which consists of three journal papers. The second part of this chapter will focus on the future plan and the potential projects, which hopefully can improve the results.6.1 Summary
Knowledge of the solar variability on various time scales is critical for better understan-ding the solar influence on the Earth’s climate system. This thesis deals with the solar variability on time scales of centuries and millennia. One of the quantities of interest is the solar irradiance. It has been measured for the last four solar cycles, but this is too short for climate studies. Thus, reconstructions of solar irradiance on longer time scales are needed. This requires long and reliable data and appropriate techniques to reconstruct the irradiance therefrom.
In Chap.1, we have first introduced the background of the solar structure, the mecha-nism of solar irradiance, the overview of the measurements and the models to reconstruct solar irradiance. Among all the existing models, SATIRE model is the most success-ful one. Two of the SATIRE versions, SATIRE-S (Yeo et al. 2014b) and SATIRE-3D (Yeo et al. 2017b), have been amazingly successful in reproducing the observed solar variability, and shown the solar variability on time scales longer than one day is indeed mainly due to the evolution of the surface magnetism. In this thesis, we will use other two SATIRE versions, SATIRE-T (Krivova et al. 2007, 2010) and SATIRE-M (Vieira et al.
2011; Wu et al. 2018a), to reconstruct the solar irradiance on centennial and millennial time scales, respectively. The SATIRE-T model uses the sunspot number as a proxy of so-lar activity to deduce the emergence rates of the active and ephemeral regions (ARs/ERs).
The evolution of solar surface magnetism is then described by a set of ordinary diff eren-tial equations (ODEs,Solanki et al. 2000,2002b). The free parameters of the model are constrained by comparing the modelled results to the corresponding reference data sets, using a genetic algorithm PIKAIA. The SATIRE-M model takes the coarser temporal
The evolution of solar surface magnetism is then described by a set of ordinary diff eren-tial equations (ODEs,Solanki et al. 2000,2002b). The free parameters of the model are constrained by comparing the modelled results to the corresponding reference data sets, using a genetic algorithm PIKAIA. The SATIRE-M model takes the coarser temporal